s_curl(3) curl-like operator for the Stokes stream function computation


form(const space M, const space& V, "s_curl");


Assembly the form associated to the s_curl operator on a finite element space V:

      b(xi,u) = |  u.s_curl(xi) dx
                / Omega

The M and V space may be a either P1 or P2 finite element space. The M space is scalar-valued while the V is vector-valued. See also form(2) and space(2).

For cartesian coordinate systems, this form coincide with the usual "curl" one (see curl(3)). In the axisymmetric case:

                | (d xi      d xi   )
      b(xi,u) = | (---- ur - ---- uz) r dr dz
                | (d z       d r    )
                / Omega

The b form is denoted as "s_curl", for Stokes stream function computation (see s_grad_grad(3)) as it is closely related to the "curl" operator (see curl(3)), but differs by the r and 1/r factors, as:

                   (       d (r xi)     d xi )
        curl(xi) = ( (1/r) -------- ; - -----)
                   (         d r        d z  )


                   ( d xi       d xi )
      s_curl(xi) = ( ----  ;  - ---- )
                   ( d r        d z  )

Notice also that the differentiation is performed on the xi variable here: b(xi,u)=(s_curl(xi),u) while the "curl" form brings the differentiation on the u vector-valued variable: (curl(u),xi), i.e. a transpose formulation.


The (r,theta,z) coordinate system has positive orientation, thus (z,r,theta) and (z,r) are positive also. But (r,z,theta) and (r,z) are negative : the sign of s_curl is then inverted to obtain the same result as if (z,r) was used.


The following piece of code build the form associated to the P1 approximation:

        geo g("square");
        space M(g, "P1");
        space V(g, "P1", "vector");
        form a(M, V, "s_curl");